Chapter 3, Problem 2AC

The metric unit of a joule (J) is a unit of _

a. potential energy.

b. work.

c. kinetic energy.

d. any of the above.

To determine

The parameter which has unit of Joule, from the following options.

Potential energy

Work

Kinetic energy

Any of the above

Answer to Problem 2AC

Solution:

Option (d) is correct.

Explanation of Solution

Introduction:

According to the concept of mechanical work, net work is done by an external force, if there is a net displacement in the same direction in which the force is applied.

Mathematically, work can be defined as:

$\begin{array}{c}W=\overrightarrow{F}\cdot \overrightarrow{d}\\ =\left|\overrightarrow{F}\right|\left|\overrightarrow{d}\right|\cos \theta \end{array}$

Here, $W$ is the work, $\overrightarrow{F}$ is the force, $\overrightarrow{d}$ is the displacement and $\theta$ is the angle between the force and the displacement.

The metric unit for Work is $\text{N}\cdot \text{m or Joule}$.

The kinetic energy of a body is the energy possessed by a body due to its motion. Example: Energy by a moving car, flowing water.

Mathematically, it is expressed as:

$K.E.=\frac{1}{2}m{v}^2$

Here, $m\text{ and }v$ are the mass and velocity of the body respectively.

The potential energy of a body is the energy possessed by a body due to its relative position. Ex: A car on a high mountain possesses potential energy due to the suitable height from the ground level.

Mathematically, it is expressed as:

$P.E.=mgh$

Here, $m$ is the mass of the body, $g$ is the acceleration due to gravity and $h$ is the height attained by the mass.

Explanation:

Reason for the correct option:

The unit for kinetic energy can be derived as follows:

$K.E=\frac{1}{2}m{v}^2$

Consider mass $m$ as $1\text{ kg}$ and velocity $v$ as $1\text{ m/s}$.

Substitute $1\text{ kg}$ for $m$ and $1\text{ m/s}$ for $v$.

$\begin{array}{c}K.E.=\frac{1}{2}\left(1\text{ kg}\right){\left(1\text{ m/s}\right)}^2\\ =\frac{1}{2}\left(\text{kg}\right)\left({\text{m}}^2{\text{/s}}^2\right)\\ =\frac{1}{2}\left(\text{J}\right)\end{array}$

Hence, Joule is a unit for kinetic energy.

Consider the expression for work.

$\begin{array}{c}W=\overrightarrow{F}\cdot \overrightarrow{d}\\ =\left|\overrightarrow{F}\right|\left|\overrightarrow{d}\right|\cos \theta \end{array}$

Consider the force as $1\text{ N}$, the displacement as $1\text{ m}$ and the angle between the force and the displacement be $0°$.

Substitute $1\text{ N}$ for $\left|\overrightarrow{F}\right|$, $1\text{ m}$ for $\left|\overrightarrow{d}\right|$ and $0°$ for $\theta$.

$\begin{array}{c}W=\left(1\text{ N}\right)\left(1\text{ m}\right)\cos 0°\\ =\left(1{\text{ kgm/s}}^2\right)\left(1\text{ m}\right)\left(1\right)\\ =1{\text{ kgm}}^2{\text{/s}}^2\\ =1\text{ J}\end{array}$

Hence, Joule is a unit for work.

The unit for potential energy can be derived by

$\begin{array}{c}P.E.=mgh\\ =\left(1\text{ kg}\right)\cdot {\text{(1 ms}}^{-2})\cdot \left(\text{1 m}\right)\\ =1\text{ kg}\cdot {\text{ms}}^{-2}\cdot \text{m}\end{array}$

Since $\text{1N}=1\text{ kg}\cdot {\text{ms}}^{-2}$. Therefore,

$\begin{array}{c}P.E.=1\text{N}\cdot \text{m}\\ =1\text{ J }\end{array}$

Joule is also the unit for potential energy.

So all options (a), (b) and (c) are true therefore option (d) is the correct choice.

Hence, option (d) is correct.

Justify reasons for the incorrect statements:

Since all options (a), (b) and (c) are true therefore option (d) is the correct choice.

Hence, option (a), (b) and (c)are incorrect choice.

Conclusion:

Joule is unit for Work, Kinetic energy and Potential Energy.